1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews / / Lecture 7 - Microecon Recap

and Discussion - “willingness to pay”  Survey of students of WTP for beer  How much for 1 beer? 2 beers? Etc.  Does similar form hold for all goods?  What types of goods different?  Economists also refer to this as demand

and (Individual) Demand Curves  Downward Sloping is a result of diminishing marginal utility of each additional unit (also consider as WTP)  Presumes that at some point you have enough to make you happy and do not value additional units Price Quantity P* Q* A B Actually an inverse demand curve (where P = f(Q) instead).

and Market Demand Price P* Q A B  If above graphs show two (groups of) consumer demands, what is social demand curve? P* Q A B

and Market Demand  Found by calculating the horizontal sum of individual demand curves  Market demand then measures ‘total consumer surplus of entire market’ P* Q

and Social WTP (i.e. market demand) Price Quantity P* Q* A B  ‘Aggregate’ demand function: how all potential consumers in society value the good or service (i.e., someone willing to pay every price…)  This is the kind of demand curves we care about

and First: Elasticities of Demand  Measurement of how “responsive” demand is to some change in price or income.  Slope of demand curve =  p/  q.  Elasticity of demand, , is defined to be the percent change in quantity divided by the percent change in price.

and Elasticities of Demand Elastic demand:  > 1. If P inc. by 1%, demand dec. by more than 1%. Unit elasticity:  = 1. If P inc. by 1%, demand dec. by 1%. Inelastic demand:  < 1 If P inc. by 1%, demand dec. by less than 1%. Q P Q P

and Elasticities of Demand Q P Q P Perfectly Inelastic Perfectly Elastic A change in price causes Demand to go to zero (no easy examples) Necessities, demand is Completely insensitive To price

and Elasticity - Some Formulas  Point elasticity = dq/dp * (p/q)  For linear curve, q = (p-a)/b so dq/dp = 1/b  Linear curve point elasticity =(1/b) *p/q = (1/b)*(a+bq)/q =(a/bq) + 1

and Sorta Timely Analysis zHow sensitive is gasoline demand to price changes? zHistorically, we have seen relatively little change in demand. Recently? zNew AAA report: higher gasoline prices have caused a 3 percent reduction in demand from a year ago.  What was  p?  q?  ? zWhat does that tell us about gasoline?

and Maglev System Example  Maglev - downtown, tech center, UPMC, CMU  20,000 riders per day forecast by developers.  Let’s assume:  price elasticity -0.3;  linear demand;  20,000 average fare of $  Estimate Total Willingness to Pay.

and Example calculations  We have one point on demand curve:  1.2 = a + b*(20,000)  We know an elasticity value:  elasticity for linear curve = 1 + a/bq  -0.3 = 1 + a/b*(20,000)  Solve with two simultaneous equations:  a = 5.2  b = or 2.0 x 10^-4

and Types of Costs - from 3-03 zPrivate - paid by consumers zSocial - paid by all of society zOpportunity - cost of foregone options zFixed - do not vary with usage zVariable - vary directly with usage zExternal - imposed by users on non-users ye.g. traffic, pollution, health risks yPrivate decisions usually ignore external

and Making Cost Functions zFundamental to analysis and policies zThree stages: y Technical knowledge of alternatives y Apply input (material) prices to options y Relate price to cost zObvious need for engineering/economics zMain point: consider cost of all parties zIncluded: labor, materials, hazard costs

and Functional Forms  TC(q) = F+ VC(q)  Use TC eq’n to generate unit costs  Average Total: ATC = TC/q  Variable: AVC = VC/q  Marginal: MC =  [TC]/  q =  TC  q  but  F/  q = 0, so MC =  [VC]/  q

and Short Run vs. Long Run Cost  Short term / short run - some costs fixed  In long run, “all costs variable”  Difference is in ‘degree of control of plans’  Generally say we are ‘constrained in the short run but not the long run’  So TC(q) < = SRTC(q)

and Firm Production Functions MC Q P What do marginal, Average cost curves Tell us? AVC Variable cost shows Non-fixed components Of producing the good Marginal costs show us Cost of producing one Additional good Where would firm produce?

and BCA Part 2: Cost Welfare Economics Continued The upper segment of a firm’s marginal cost curve corresponds to the firm’s SR supply curve. Again, diminishing returns occur. Quantity Price Supply=MC At any given price, determines how much output to produce to maximize profit AVC

and Supply/Marginal Cost Notes Quantity Price Supply=MC At any given price, determines how much output to produce to maximize profit P* Q1 Q* Q2 Demand: WTP for each additional unit Supply: cost incurred for each additional unit

and Supply/Marginal Cost Notes Quantity Price Supply=MC Area under MC is TVC - why? P* Q1 Q* Q2 Recall: We always want to be considering opportunity costs (total asset value to society) and not accounting costs

and Unifying Cost and Supply  Economists learn “Supply and Demand”  Equilibrium (meeting point): where S = D  In our case, substitute ‘cost’ for supply  Why cost? Need to trade-off Demand  Using MC is a standard method  Recall this is a perfectly competitive world!

and Example  Demand Function: p = 4 - 3q  Supply function: p = 1.5q  Assume equilibrium, what is p,q?  In eq: S=D; 4-3q=1.5q ; 4.5q=4 ; q=8/9  P=1.5q=(3/2)*(8/9)= 4/3

and Pricing Strategies  Highway pricing  If price set equal to AC (which is assumed to be TC/q then at q, total costs covered  p ~ AVC: manages usage of highway  p = f(fares, fees, travel times, discomfort)  Price increase=> less users (BCA)  MC pricing: more users, higher price  What about social/external costs?  Might want to set p=MSC

and Estimating Linear Demand Functions zAs above, sometimes we don’t know demand zFocus on demand (care more about CS) but can use similar methods to estimate costs (supply) zOrdinary least squares regression used yminimize the sum of squared deviations between estimated line and p,q observations: p = a + bq + e yStandard algorithms to compute parameter estimates - spreadsheets, Minitab, S, etc. yEstimates of uncertainty of estimates are obtained (based upon assumption of identically normally distributed error terms). zCan have multiple linear terms

and Log-linear Function zq = a(p) b (hh) c ….. zConditions: a positive, b negative, c positive,... zIf q = a(p) b : Elasticity interesting = (dq/dp)*(p/q) = abp (b-1) *(p/q) = b*(ap b /ap b ) = b. yConstant elasticity at all points. zEasiest way to estimate: linearize and use ordinary least squares regression (see Chap 12) yE.g., ln q = ln a + b ln(p) + c ln(hh)..

and Log-linear Function  q = a*p b and taking log of each side gives: ln q = ln a + b ln p which can be re-written as q’ = a’ + b p’, linear in the parameters and amenable to OLS regression.  This violates error term assumptions of OLS regression.  Alternative is maximum likelihood - select parameters to max. chance of seeing obs.

and Maglev Log-Linear Function  q = ap b - From above, b = -0.3, so if p = 1.2 and q = 20,000; so 20,000 = a*(1.2) -0.3 ; a = 21,124.  If p becomes 1.0 then q = 21,124*(1) -0.3 = 21,124.  Linear model - 21,000  Remaining revenue, TWtP values similar but NOT EQUAL.

and Demand Example (cont)  Maglev Demand Function:  p = *q  Revenue: 1.2*20,000 = $ 24,000 per day  TWtP = Revenue + Consumer Surplus  TWtP = pq + 1/2*(a-p)q = 1.2*20, *( )*20,000 = 24, ,000 = $ 64,000 per day.

and Change in Fare to $ 1.00  From demand curve: 1.0 = q, so q becomes 21,000.  Using elasticity: 16.7% fare change (1.2-1/1.2), so q would change by -0.3*16.7 = 5.001% to 21,002 (slightly different value)  Change to Revenue = 1*21, *20,000 = 21, ,000 = -3,000.  Change CS = 0.5*(0.2)*(20,000+21,000)= 4,100  Change to TWtP = (21,000-20,000)*1 + (1.2-1)*(21, ,000)/2 = 1,100.